Subset Relation

A Subset Relation is a binary set relation between sets ([math]\displaystyle{ X_1,X_2 }[/math]) that is True If [math]\displaystyle{ \forall x \in X_1: x \in X_2 }[/math].

• AKA: SubsetOf Operation, Inclusion Relation, ⊆.
• Example(s):
  • {} ⊆ {1, 2, 3}.
  • {1, 2, 3} ⊆ {1, 2, 3}.
  • {1, 2} ⊆ {1, 2, 3}, also a Proper Subset Relation(⊂)
  • a Subclass Relation.
  • a Population Subset.
  • …
• Counter-Example(s):
  • a Not Subset Relation (⊄), such as {A} ⊄ {1, 2, 3}.
  • Superset.
• See: Set of Subsets, Empty Set.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Subset#Definitions Retrieved:2017-6-8.
  • If A and B are sets and every element of A is also an element of B, then:

     :* A is a subset of (or is included in) B, denoted by [math]\displaystyle{ A \subseteq B }[/math] ,

    :or equivalently

     :* B is a superset of (or includes) A, denoted by [math]\displaystyle{ B \supseteq A. }[/math] If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then

     :* A is also a proper (or strict) subset of B ; this is written as [math]\displaystyle{ A \subsetneq B. }[/math] :or equivalently

     :* B is a proper superset of A ; this is written as [math]\displaystyle{ B \supsetneq A. }[/math] For any set S, the inclusion relation ⊆ is a partial order on the set [math]\displaystyle{ \mathcal{P}(S) }[/math] of all subsets of S (the power set of S) defined by [math]\displaystyle{ A \leq B \iff A \subseteq B }[/math] . We may also partially order [math]\displaystyle{ \mathcal{P}(S) }[/math] by reverse set inclusion by defining [math]\displaystyle{ A \leq B \iff B \subseteq A }[/math] .

    When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.

2009

• (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
  • a set whose members are members of another set; a set contained within another set

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